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Theorem mulsrpr 9344
Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
mulsrpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )

Proof of Theorem mulsrpr
Dummy variables  x  y  z  w  v  u  t  s  f 
g  h  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4654 . 2  |-  <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>.  e.  _V
2 opex 4654 . 2  |-  <. (
( a  .P.  g
)  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  e.  _V
3 opex 4654 . 2  |-  <. (
( c  .P.  t
)  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >.  e.  _V
4 enrex 9338 . 2  |-  ~R  e.  _V
5 enrer 9336 . 2  |-  ~R  Er  ( P.  X.  P. )
6 df-enr 9327 . 2  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
7 oveq12 6199 . . . 4  |-  ( ( z  =  a  /\  u  =  d )  ->  ( z  +P.  u
)  =  ( a  +P.  d ) )
8 oveq12 6199 . . . 4  |-  ( ( w  =  b  /\  v  =  c )  ->  ( w  +P.  v
)  =  ( b  +P.  c ) )
97, 8eqeqan12d 2474 . . 3  |-  ( ( ( z  =  a  /\  u  =  d )  /\  ( w  =  b  /\  v  =  c ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
109an42s 823 . 2  |-  ( ( ( z  =  a  /\  w  =  b )  /\  ( v  =  c  /\  u  =  d ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( a  +P.  d )  =  ( b  +P.  c ) ) )
11 oveq12 6199 . . . 4  |-  ( ( z  =  g  /\  u  =  s )  ->  ( z  +P.  u
)  =  ( g  +P.  s ) )
12 oveq12 6199 . . . 4  |-  ( ( w  =  h  /\  v  =  t )  ->  ( w  +P.  v
)  =  ( h  +P.  t ) )
1311, 12eqeqan12d 2474 . . 3  |-  ( ( ( z  =  g  /\  u  =  s )  /\  ( w  =  h  /\  v  =  t ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
1413an42s 823 . 2  |-  ( ( ( z  =  g  /\  w  =  h )  /\  ( v  =  t  /\  u  =  s ) )  ->  ( ( z  +P.  u )  =  ( w  +P.  v
)  <->  ( g  +P.  s )  =  ( h  +P.  t ) ) )
15 df-mpr 9326 . 2  |-  .pR  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .P.  u
)  +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  (
v  .P.  u )
) >. ) ) }
16 oveq12 6199 . . . . 5  |-  ( ( w  =  a  /\  u  =  g )  ->  ( w  .P.  u
)  =  ( a  .P.  g ) )
17 oveq12 6199 . . . . 5  |-  ( ( v  =  b  /\  f  =  h )  ->  ( v  .P.  f
)  =  ( b  .P.  h ) )
1816, 17oveqan12d 6209 . . . 4  |-  ( ( ( w  =  a  /\  u  =  g )  /\  ( v  =  b  /\  f  =  h ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) )
1918an4s 822 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) )
20 oveq12 6199 . . . . 5  |-  ( ( w  =  a  /\  f  =  h )  ->  ( w  .P.  f
)  =  ( a  .P.  h ) )
21 oveq12 6199 . . . . 5  |-  ( ( v  =  b  /\  u  =  g )  ->  ( v  .P.  u
)  =  ( b  .P.  g ) )
2220, 21oveqan12d 6209 . . . 4  |-  ( ( ( w  =  a  /\  f  =  h )  /\  ( v  =  b  /\  u  =  g ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( a  .P.  h
)  +P.  ( b  .P.  g ) ) )
2322an42s 823 . . 3  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( a  .P.  h )  +P.  ( b  .P.  g ) ) )
2419, 23opeq12d 4165 . 2  |-  ( ( ( w  =  a  /\  v  =  b )  /\  ( u  =  g  /\  f  =  h ) )  ->  <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( a  .P.  g )  +P.  ( b  .P.  h
) ) ,  ( ( a  .P.  h
)  +P.  ( b  .P.  g ) ) >.
)
25 oveq12 6199 . . . . 5  |-  ( ( w  =  c  /\  u  =  t )  ->  ( w  .P.  u
)  =  ( c  .P.  t ) )
26 oveq12 6199 . . . . 5  |-  ( ( v  =  d  /\  f  =  s )  ->  ( v  .P.  f
)  =  ( d  .P.  s ) )
2725, 26oveqan12d 6209 . . . 4  |-  ( ( ( w  =  c  /\  u  =  t )  /\  ( v  =  d  /\  f  =  s ) )  ->  ( ( w  .P.  u )  +P.  ( v  .P.  f
) )  =  ( ( c  .P.  t
)  +P.  ( d  .P.  s ) ) )
2827an4s 822 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( ( w  .P.  u )  +P.  ( v  .P.  f
) )  =  ( ( c  .P.  t
)  +P.  ( d  .P.  s ) ) )
29 oveq12 6199 . . . . 5  |-  ( ( w  =  c  /\  f  =  s )  ->  ( w  .P.  f
)  =  ( c  .P.  s ) )
30 oveq12 6199 . . . . 5  |-  ( ( v  =  d  /\  u  =  t )  ->  ( v  .P.  u
)  =  ( d  .P.  t ) )
3129, 30oveqan12d 6209 . . . 4  |-  ( ( ( w  =  c  /\  f  =  s )  /\  ( v  =  d  /\  u  =  t ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) )
3231an42s 823 . . 3  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  ( ( w  .P.  f )  +P.  ( v  .P.  u
) )  =  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) )
3328, 32opeq12d 4165 . 2  |-  ( ( ( w  =  c  /\  v  =  d )  /\  ( u  =  t  /\  f  =  s ) )  ->  <. ( ( w  .P.  u )  +P.  ( v  .P.  f
) ) ,  ( ( w  .P.  f
)  +P.  ( v  .P.  u ) ) >.  =  <. ( ( c  .P.  t )  +P.  ( d  .P.  s
) ) ,  ( ( c  .P.  s
)  +P.  ( d  .P.  t ) ) >.
)
34 oveq12 6199 . . . . 5  |-  ( ( w  =  A  /\  u  =  C )  ->  ( w  .P.  u
)  =  ( A  .P.  C ) )
35 oveq12 6199 . . . . 5  |-  ( ( v  =  B  /\  f  =  D )  ->  ( v  .P.  f
)  =  ( B  .P.  D ) )
3634, 35oveqan12d 6209 . . . 4  |-  ( ( ( w  =  A  /\  u  =  C )  /\  ( v  =  B  /\  f  =  D ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) )
3736an4s 822 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( ( w  .P.  u )  +P.  (
v  .P.  f )
)  =  ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) )
38 oveq12 6199 . . . . 5  |-  ( ( w  =  A  /\  f  =  D )  ->  ( w  .P.  f
)  =  ( A  .P.  D ) )
39 oveq12 6199 . . . . 5  |-  ( ( v  =  B  /\  u  =  C )  ->  ( v  .P.  u
)  =  ( B  .P.  C ) )
4038, 39oveqan12d 6209 . . . 4  |-  ( ( ( w  =  A  /\  f  =  D )  /\  ( v  =  B  /\  u  =  C ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( A  .P.  D )  +P.  ( B  .P.  C ) ) )
4140an42s 823 . . 3  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( ( w  .P.  f )  +P.  (
v  .P.  u )
)  =  ( ( A  .P.  D )  +P.  ( B  .P.  C ) ) )
4237, 41opeq12d 4165 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( ( w  .P.  u )  +P.  (
v  .P.  f )
) ,  ( ( w  .P.  f )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >.
)
43 df-mr 9330 . 2  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. a E. b E. c E. d ( ( x  =  [ <. a ,  b >. ]  ~R  /\  y  =  [ <. c ,  d >. ]  ~R  )  /\  z  =  [
( <. a ,  b
>.  .pR  <. c ,  d
>. ) ]  ~R  )
) }
44 df-nr 9328 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
45 mulcmpblnr 9342 . 2  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  (
c  e.  P.  /\  d  e.  P. )
)  /\  ( (
g  e.  P.  /\  h  e.  P. )  /\  ( t  e.  P.  /\  s  e.  P. )
) )  ->  (
( ( a  +P.  d )  =  ( b  +P.  c )  /\  ( g  +P.  s )  =  ( h  +P.  t ) )  ->  <. ( ( a  .P.  g )  +P.  ( b  .P.  h ) ) ,  ( ( a  .P.  h )  +P.  (
b  .P.  g )
) >.  ~R  <. ( ( c  .P.  t )  +P.  ( d  .P.  s ) ) ,  ( ( c  .P.  s )  +P.  (
d  .P.  t )
) >. ) )
461, 2, 3, 4, 5, 6, 10, 14, 15, 24, 33, 42, 43, 44, 45ovec 7310 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3981  (class class class)co 6190   [cec 7199   P.cnp 9127    +P. cpp 9129    .P. cmp 9130    .pR cmpr 9133    ~R cer 9134   R.cnr 9135    .R cmr 9140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-omul 7025  df-er 7201  df-ec 7203  df-qs 7207  df-ni 9142  df-pli 9143  df-mi 9144  df-lti 9145  df-plpq 9178  df-mpq 9179  df-ltpq 9180  df-enq 9181  df-nq 9182  df-erq 9183  df-plq 9184  df-mq 9185  df-1nq 9186  df-rq 9187  df-ltnq 9188  df-np 9251  df-plp 9253  df-mp 9254  df-ltp 9255  df-mpr 9326  df-enr 9327  df-nr 9328  df-mr 9330
This theorem is referenced by:  mulclsr  9352  mulcomsr  9357  mulasssr  9358  distrsr  9359  m1m1sr  9361  1idsr  9366  00sr  9367  recexsrlem  9371  mulgt0sr  9373
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